Upper bounds for partial spreads

UPPER BOUNDS FOR PARTIAL SPREADS

SASCHA KURZ^?

ABSTRACT. Apartialt-spreadinFⁿq is a collection oft-dimensional subspaces with trivial intersection such that each non-zero vector is covered at most once. We present some improved upper bounds on the maximum sizes.

Keywords:Galois geometry, partial spreads, constant dimension codes, and vector space partitions MSC:51E23; 05B15, 05B40, 11T71, 94B25

1. INTRODUCTION

Let q > 1 be a prime power andn a positive integer. Avector space partitionP of Fⁿq is a collection of subspaces with the property that every non-zero vector is contained in a unique member ofP. IfPcontainsmd

subspaces of dimensiond, thenP is of typek^mk. . .1^m1. We may leave out some of the cases withmd = 0.

Subspaces of dimension1are calledholes. If there is at least one non-hole, thenPis called non-trivial.

Apartialt-spreadinFⁿq is a collection oft-dimensional subspaces such that the non-zero vectors are covered at most once, i.e., a vector space partition of typet^mt1^m1. ByAq(n,2t;t)we denote the maximum value of mt1. Writingn = kt+r, withk, r ∈ N0andr ≤t−1, we can state that forr ≤ 1orn ≤ 2tthe exact value ofAq(n,2t;t)was known for more than forty years [1]. Via a computer search the casesA2(3k+ 2,6; 3) were settled in 2010 by El-Zanati et al. [5]. In 2015 the caseq=r= 2was resolved by continuing the original approach of Beutelspacher [13], i.e., byconsideringthe set of holes in(n−2)-dimensional subspaces and some averaging arguments. Very recently, N˘astase and Sissokho found a very clear generalized averaging method for the number of holes in(n−j)-dimensional subspaces, wherej ≤ t−2, and generalq, see [14]. Their Theorem 5 determines the exact values of A_q(kt+r,2t;t)in all cases wheret > r

1

q := ^q_q−1^r−1. Here, we streamline and generalize their approach leading to improved upper bounds onAq(n,2t;t), c.f. [15].

2. SUBSPACES WITH THE MINIMUM NUMBER OF HOLES

Definition 2.1. A vector space partitionP ofFⁿq hashole-type(t, s, m₁), if it is of typet^mt. . . s^ms1^m1, for some integersn > t≥s≥2,mi∈N0fori∈ {1, s, . . . , t}, andPis non-trivial.

Lemma 2.2. (C.f.[14, Proof of Lemma 9].) LetP be a non-trivial vector space partition ofFⁿq of hole-type (t, s, m1)andl, x∈N⁰withPt

i=smi=lq^s+x.PH={U∩H : U ∈ P}is a vector space partition of type t^m0t. . .(s−1)^m0s−11^m01, for a hyperplaneH withmb1holes (ofP). We havemb1 ≡ ^m1+x−1_q (modq^s−1). If s >2, thenPHis non-trivial andm⁰₁=mb₁.

PROOF. IfU ∈ P, thendim(U)−dim(U∩H)∈ {0,1}for an arbitrary hyperplaneH. SinceP is non-trivial, we haven≥s. Fors >2, counting the1-dimensional subspaces ofFⁿq andH, viaPandPH, yields

(lq^s+x)· s

1

q

+aq^s+m₁= n

1

q

and (lq^s+x)· s−1

1

q

+a⁰q^s−1+mb₁= n−1

1

q

for somea, a⁰∈N0. Since1 +q·n−1 1

q−n 1

q = 0we conclude1 +qmb₁−m₁−x≡0 (modq^s). Thus, Z3mb1≡^m1+x−1_q (mod q^s−1). Fors= 2we have

lq²+x

·(q+ 1) +aq²+m1= n

1

q

and lq²+x

·1 +a⁰q+mb1= n−1

1

q

?The work of the author was supported by the ICT COST Action IC1104 and grant KU 2430/3-1 – Integer Linear Programming Models for Subspace Codes and Finite Geometry from the German Research Foundation.

1The more general notationAq(n,2t−2w;t)denotes the maximum cardinality of a collection oft-dimensional subspaces, whose pair- wise intersections have a dimension of at mostw. Those objects are calledconstant dimension codes, see e.g. [6]. For known bounds, we re- fer tohttp://subspacecodes.uni-bayreuth.de[10] containing also the generalization tosubspace codesof mixed dimension.

1

2 SASCHA KURZ

leading to the same conclusionmb1≡ ^m1+x−1_q (modq^s−1).

Lemma 2.3. (C.f.[14, Proof of Lemma 9].) LetP be a vector space partition ofFⁿq of hole-type(t, s, m₁), l, x ∈ N0 withPt

i=sm_i = lq^s+x, andb, c ∈ Zwithm₁ = bq^s+c ≥ 1. Ifx ≥ 1, then there exists a hyperplaneHb withmb1=bbq^s−1+bcholes, wherebc:= ^c+x−1_q ∈Zandb >bb∈Z.

PROOF. Apply Lemma 2.2 and observem₁≡c (modq^s). Let the number of holes inHb be minimal. Then,

mb1≤average number of holes per hyperplane=m1· n−1

1

q

/ n

1

q

< m1

q . (1)

Assumingbb≥byieldsqmb₁≥q·(bq^s−1+bc) =bq^s+c+x−1≥m₁, which contradicts Inequality (1).

Corollary 2.4. Using the notation from Lemma 2.3, letP be a non-trivial vector space partition withx≥1 andf be the largest integer such thatq^f dividesc. For each0≤j≤s−max{1, f}there exists an(n−j)- dimensional subspaceU containingmb₁holes withmb₁≡bc (modq^s−j)andmb₁≤(b−j)·q^s−j+bc, where bc=^c+[^j1]_q^·(x−1)

q^j .

Proof. Observemb₁≡c6≡0 (modq^s−j), i.e.,mb₁≥1, for allj < s−f. Lemma 2.5. LetP be a non-trivial vector space partition of typet^mt1^m1 ofFⁿq withm_t = lq^t+x, where l = ^qn−t_qt−1^−qr, x ≥ 2, t = r

1

q + 1−z+u > r, q^f|x−1, q^f+1 - x−1, and f, u, z, r, x ∈ N0. For max{1, f} ≤y≤tthere exists a(n−t+y)-dimensional subspaceU withL≤(z+y−1)q^y+wholes, where w=−(x−1)y

1

qandL≡w(mod q^y).

PROOF. Apply Corollary 2.4 withs=t,j =t−y,b=r 1

q, andm₁=r 1

qq^t−t 1

q(x−1).

Lemma 2.6. LetPbe a vector space partition ofFⁿq withc≥1holes andaidenote the number of hyperplanes containingiholes. Then,Pc

i=0ai =n 1

q,Pc

i=0iai=c·_n−1

1

qandPc

i=0i(i−1)ai=c(c−1)·_n−2

1

q. PROOF. Double-count the incidences of the tuples(H),(B1, H), and(B1, B2, H), whereH is a hyperplane

andB16=B2are points contained inH.

Lemma 2.7. Let ∆ = q^s−1,m ∈ Z, andP be a vector space partition ofFⁿq of hole-type (t, s, c). Then, τq(c,∆, m)·^q_∆ⁿ⁻²2 −m(m−1)≥0, where

τ_q(c,∆, m) =m(m−1)∆²q²−c(2m−1)(q−1)∆q+c(q−1)

c(q−1) + 1 . PROOF. Consider the three equations from Lemma 2.6. (c−m∆)

c−(m−1)∆

times the first minus

2c−(2m−1)∆−1

times the second plus the third equation, and then divided by∆²/(q−1), gives

(q−1)·

bc/∆c

X

h=0

(m−h)(m−h−1)a_c−h∆=τq(c,∆, m)·qⁿ⁻²

∆² −m(m−1)

due to Lemma 2.2. Finally, we observeai≥0and(m−h)(m−h−1)≥0for allm, h∈Z. Lemma 2.8. For integersn > t≥s≥2and1≤i≤s−1, there exists no vector space partitionPofFⁿq of hole-type(t, s, c), wherec=i·q^s−s

1

q+s−1.²

PROOF. Assume the contrary and apply Lemma 2.7 withm=i(q−1). Settingy =s−1−iand∆ =q^s−1 we compute

τq(c,∆, m) = −q∆(y(q−1) + 2) + (s−1)²q²−q(s−1)(2s−5) + (s−2)(s−3).

Usingy≥0we obtainτ₂(c,∆, m)≤s²+s−2^s+1<0. Fors= 2, we haveτ_q(c,∆, m) =−q²+q <0and forq, s >2we haveτ_q(c,∆, m)≤ −2q^s+ (s−1)²q²<0. Thus, Lemma 2.7 yields a contradiction.

2For more general non-existence results of vector space partitions see e.g. [9, Theorem 1] and the related literature.

UPPER BOUNDS FOR PARTIAL SPREADS 3

Theorem 2.9. (C.f. [14, Lemma 10].) For integersr ≥ 1,k ≥ 2, u ≥ 0, and0 ≤ z ≤ r 1

q/2 witht = r

1

q+ 1−z+u > rwe haveAq(n,2t;t)≤lq^t+ 1 +z(q−1), wherel= ^qn−t_qt−1^−qr andn=kt+r.

PROOF. Apply Lemma 2.5 withx= 2 +z(q−1)andy = z+ 1. Ifz = 0, thenL < 0. Forz ≥ 1, apply

Lemma 2.8. Thus,Aq(n,2t;t)≤lq^t+x−1.

The known constructions for partialt-spreads giveAq(kt+r,2t;t)≥lq^t+ 1, see e.g. [1] (or [13] for an interpretation using the more general multilevel construction for subspace codes). Thus, Theorem 2.9 is tight fort≥r

1

q+ 1, c.f. [14, Theorem 5].

Theorem 2.10. (C.f.[15, Theorem 6,7].) For integersr≥1,k≥2,y≥max{r,2},z≥0withλ=q^y,y≤t, t=r

1

q+ 1−z > r,n=kt+r, andl= ^qn−t_qt−1^−qr, we have Aq(n,2t;t)≤lq^t+

λ−1

2−1 2

p1 + 4λ(λ−(z+y−1)(q−1)−1)

. PROOF. From Lemma 2.5 we concludeL≤(z+y−1)q^y−(x−1)y

1

qandL≡ −(x−1)y 1

q (modq^y)for the number of holes of a certain(n−t+y)-dimensional subspaceUofFⁿq.P_U :={P∩U |P∈ P}is of hole- type(t, y, L)ify ≥2. Next, we will show thatτq(c,∆, m)≤0, where∆ =q^y−1andc =iq^y−(x−1)y

1

q

with1≤i≤z+y−1, for suitable integersxandm. Note that, in order to apply Lemma 2.5, we have to satisfy x≥2andy≥ffor all integersfwithq^f|x−1. Applying Lemma 2.7 then gives the desired contradiction, so thatAq(n,2t;t)≤lq^t+x−1.

We choose³m=i(q−1)−(x−1) + 1, so thatτq(c,∆, m) =x²−(2λ+ 1)x+λ(i(q−1) + 2). Solving τ_q(c,∆, m) = 0forxgivesx₀ = λ+ ¹₂ ±¹₂θ(i), whereθ(i) = p

1−4iλ(q−1) + 4λ(λ−1). We have τ_q(c,∆, m)≤0for|2x−2λ−1| ≤θ(i). We need to find an integerx≥2such that this inequality is satisfied for all1≤i≤z+y−1. The strongest restriction is attained fori=z+y−1. Sincez+y−1≤r

1

q and u=q^y ≥q^r, we haveθ(i)≥θ(z+y−1)≥1, so thatτq(c,∆, m)≤0forx=

u+¹₂−¹₂θ(z+y−1) . (Observex≤λ+¹₂+¹₂θ(z+y−1)due toθ(z+y−1)≥1.) Sincex≤λ+ 1, we havex−1≤λ=q^y, so that q^f|x−1impliesf ≤yprovidedx≥2. The latter is true due toθ(z+y−1)≤p

1−4λ(q−1) + 4λ(λ−1)≤ p1 + 4λ(λ−2)<2(λ−1), which impliesx≥3

2

= 2.

So far we have constructed a suitablem∈Zsuch thatτ_q(c,∆, m)≤0forx=

λ+¹₂−¹₂θ(z+y−1) . Ifτ_q(c,∆, m)<0, then Lemma 2.7 gives a contradiction, so that we assumeτ_q(c,∆, m) = 0in the following.

Ifi < z+y−1we haveτ_q(c,∆, m) <0 due toθ(i) > θ(z+y−1), so that we assumei =z+y−1.

Thus,θ(z+y−1)∈ N0. However, we can writeθ(z+y−1)² = 1 + 4λ(λ−(z+y−1)(q−1)−1) = (2w−1)²= 1+4w(w−1)for some integerw. Ifw /∈ {0,1}, thengcd(w, w−1) = 1, so that eitherλ=q^y|w orλ =q^y | w−1. Thus, in any case, w≥ q^y, which is impossible since(z+y−1)(q−1) ≥1. Finally, w∈ {0,1}impliesw(w−1) = 0, so thatλ−(z+y−1)(q−1)−1 = 0. Thus,z+y−1 =y

1

q ≥r 1

q

sincey≥r. The assumptionsy≤tandt=r 1

q+ 1−zimplyz+y−1 =r 1

qandy=r. This givest=r,

which is excluded.

Setting y = t in Theorem 2.10 yields [4, Corollary 8], which is based on [3, Theorem 1B]. And indeed, our analysis is very similar to the technique⁴used in [3]. Compared to [3, 4], the new ingredients essentially are lemmas 2.2 and 2.3, see also [14, Proof of Lemma 9]. [4, Corollary 8], e.g., givesA2(15,12; 6) ≤516, A2(17,14; 7) ≤ 1028, and A9(18,16; 8) ≤ 3486784442, while Theorem 2.10 gives A2(15,12; 6) ≤ 515, A2(17,14; 7)≤1026, andA9(18,16; 8)≤3486784420. Postponing the details and proofs to a more extensive and technical paper [12], we state:

• 2⁴l+ 1≤A2(4k+ 3,8; 4)≤2⁴l+ 4, wherel=^24k−1₂4−1⁻²³ andk≥2, e.g.,A2(11,8; 4)≤132;

• 2⁶l+ 1≤A2(6k+ 4,12; 6)≤2⁶l+ 8, wherel= ^26k−2₂6−1⁻²⁴ andk≥2, e.g.,A2(16,12; 6)≤1032;

• 2⁶l+ 1≤A2(6k+ 5,12; 6)≤2⁶l+ 18, wherel=^26k−1₂6−1⁻²⁵ andk≥2, e.g.,A2(17,12; 6)≤2066;

• 3⁴l+ 1≤A3(4k+ 3,8; 4)≤3⁴l+ 14, wherel= ^34k−1₃4−1⁻³³ andk≥2, e.g.,A3(11,8; 4)≤2201;

3Solving^{∂τq(c,∆,m)}_∂m = 0, i.e., minimizingτq(c,∆, m), yieldsm=i(q−1)−(x−1) +¹₂ +^x−1_qy . Fory≥rwe can assume x−1< q^ydue the known constructions for partial spreads, so that up-rounding yields the optimum integer choice. Fory < rthe interval u+¹₂−¹₂θ(i), u+¹₂+¹₂θ(i)

may contain no integer.

4Actually, their analysis grounds on [16] and is strongly related to the classical second-order Bonferroni Inequality [2, 7, 8] in Probability Theory, see e.g. [11, Section 2.5] for another application for bounds on subspace codes.

4 SASCHA KURZ

• 3⁵l+ 1≤A3(5k+ 3,10; 5)≤3⁵l+ 13, wherel=^35k−2₃3−1⁻³⁵ andk≥2, e.g.,A3(13,10; 5)≤6574;

• 3⁵l+ 1≤A₃(5k+ 4,10; 5)≤3⁵l+ 44, wherel=^35k−1₃₅₋₁⁻³⁴ andk≥2, e.g.,A₃(14,10; 5)≤19727;

• 3⁶l+ 1≤A3(6k+ 4,12; 6)≤3⁶l+ 41, wherel=^36k−2₃6−1⁻³⁴ andk≥2, e.g.,A3(16,12; 6)≤59090;

• 3⁶l+1≤A₃(6k+5,12; 6)≤3⁶l+133, wherel= ^36k−1₃₆₋₁⁻³⁵ andk≥2, e.g.,A₃(17,12; 6)≤177280;

• 3⁷l+ 1≤A3(7k+ 4,14; 7)≤3⁷l+ 40, wherel= ^37k−3₃7−1⁻³⁴ andk≥2, e.g.,A3(18,14; 7)≤177187;

• 4⁵l+ 1≤A₄(5k+ 3,10; 5)≤4⁵l+ 32, wherel=^45k−2₄5−1⁻⁴³ andk≥2, e.g.,A₄(13,10; 5)≤65568;

• 4⁶l+ 1≤A4(6k+ 3,12; 6)≤4⁶l+ 30, wherel= ^46k−3₄6−1⁻⁴³ andk≥2, e.g.,A4(15,12; 6)≤262174;

• 4⁶l+1≤A4(6k+5,12; 6)≤4⁶l+548, wherel=^46k−1₄6−1⁻⁴⁵ andk≥2, e.g.,A4(17,12; 6)≤4194852;

• 4⁷l+1≤A4(7k+4,14; 7)≤4⁷l+128, wherel=^47k−3₄7−1⁻⁴⁴ andk≥2, e.g.,A4(18,14; 7)≤4194432;

• 5⁵l+ 1≤A5(5k+ 2,10; 5)≤5⁵l+ 7, wherel= ^55k−3₅5−1⁻⁵² andk≥2, e.g.,A5(12,10; 5)≤78132;

• 5⁵l+1≤A5(5k+4,10; 5)≤5⁵l+329, wherel=^55k−1₅5−1⁻⁵⁴ andk≥2, e.g.,A5(14,10; 5)≤1953454;

• 7⁵l+ 1 ≤ A7(5k+ 4,10; 5) ≤ 7⁵l+ 1246, wherel = ^75k−1₇5−1⁻⁷² andk ≥ 2, e.g., A7(14,10; 5) ≤ 40354853;

• 8⁴l+ 1≤A8(4k+ 3,8; 4)≤8⁴l+ 264, wherel= ^84k−1₈4−1⁻⁸³ andk≥2, e.g.,A8(11,8; 4)≤2097416;

• 8⁵l+1≤A8(5k+2,10; 5)≤8⁵l+25, wherel=^85k−3₈5−1⁻⁸² andk≥2, e.g.,A8(12,10; 5)≤2097177;

• 8⁶l+1≤A8(6k+2,12; 6)≤8⁶l+21, wherel= ^86k−4₈6−1⁻⁸²andk≥2, e.g.,A8(14,12; 6)≤16777237;

• 9³l+ 1≤A9(3k+ 2,6; 3)≤9³l+ 41, wherel= ^93k−1₉3−1⁻⁹² andk≥2, e.g.,A9(8,6; 3)≤59090;

• 9⁵l+ 1 ≤ A9(5k+ 3,10; 5) ≤ 9⁵l+ 365, wherel = ^95k−2₉5−1⁻⁹³ andk ≥ 2, e.g., A9(13,10; 5) ≤ 43047086;

c.f. the web-page mentioned in footnote 1 for more numerical values and comparisons of the different upper bounds.

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DEPARTMENT OFMATHEMATICS, UNIVERSITY OFBAYREUTH, 95440 BAYREUTH, GERMANY E-mail address:sascha.kurz@uni-bayreuth.de